Light a grill. In a large pot of boiling salted water, cook the pasta until al dente. Drain the pasta, reserving 6 tablespoons of the cooking water. Return the al dente pasta to the pot.

Meanwhile, in a bowl, toss the squid with 1 tablespoon of the oil; season with salt and pepper and grill over moderately high heat, turning once, until lightly charred and white throughout, 4 minutes. Transfer the squid to a work surface. Slice the bodies crosswise into 1/4-inch-thick rings and quarter the tentacles.

In a small skillet, heat the remaining 1/4 cup of olive oil. Add the garlic and cook over moderate heat until golden, about 3 minutes. Add the crushed red pepper and cook for 30 seconds. Remove from the heat and stir in the lemon juice.

Stir the lemon-garlic sauce into the pasta. Stir in the arugula, chickpeas, squid and the reserved pasta cooking water. Cook over low heat, tossing, until the arugula is wilted, about 1 minute. Season with salt and pepper. Transfer to bowls and serve.